Homotopy theory in toric topology (Q2816814)

In toric topology two important spaces, the Davis-Januszkiewicz space \(DJ_K\) and the moment-angle complex \({\mathcal{L}}_K\), are associated to each simplicial complex \(K\). These spaces and their generalizations have applications to algebra, geometry and combinatorics. In the first part of the paper, the authors survey some of the main results in the homotopy theory of moment-angle complexes and their generalization to polyhedral products.NEWLINENEWLINEWhen \(K\) is a simplicial complex on \(m\) vertices, there is a homotopy fibration NEWLINE\[NEWLINE {\mathcal{L}}_K \overset{\tilde{w}}\longrightarrow DJ_K\longrightarrow \prod_{i=1}^m{\mathbb{C}}P^\infty. NEWLINE\]NEWLINE In the second part of the paper, the authors study the map \(\tilde{w}\) and show how the homotopy theoretic information about the moment-angle complex \({\mathcal{L}}_K\) can be related to the Davis-Januszkiewicz space \(DJ_K\). They show that for a certain family of simplicial complexes \(K\), \({\mathcal{L}}_K\) is homotopy equivalent to a wedge of spheres. The homotopy equivalence can be chosen so that the map \(\tilde{w}\) consists of a specified collection of higher and iterated Whitehead products. The authors expect that their results can be generalized to a much larger family of simplicial complexes and also to the map \({\mathcal{L}}_K({\underline{X}})\to DJ_K({\underline{X}})\) between polyhedral products.